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<article class="li"><h3 class="heading">
<span class="type">Item</span><span class="space"> </span><span class="codenumber">2</span><span class="period">.</span>
</h3>
<p>For the system of ODEs</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
\begin{aligned}
\frac{dy}{dt}-\frac{dx}{dt}+y +2x &amp;=&amp;e^t \label{odesys1} \\
\frac{dy}{dt}+\frac{dx}{dt}-x &amp;=&amp;e^{2t} \\
{\rm Initial \ data:} \ \ \ x(0),y(0)&amp;=&amp;1, \label{odesys}
\end{aligned}
\end{equation*}
</div>
<ol class="lower-alpha">
<li>
<p>transform to obtain $$</p>
<p>$$</p>
</li>
<li>
<p>Rearranging, $$</p>
<p>$$</p>
</li>
<li>
<p>To eliminate <span class="process-math">\(Y(s),\)</span> multiply (<code class="code-inline tex2jax_ignore">[cross-reference to target(s) "s1" missing or not unique]</code>) by <span class="process-math">\(s\)</span> and (<code class="code-inline tex2jax_ignore">[cross-reference to target(s) "s2" missing or not unique]</code>) by <span class="process-math">\((s+1)\)</span> then subtract, and deduce as follows $$</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
Then, by partial fractions,
\end{equation*}
</div>
<p class="continuation">X(s) = +  -  -  .$$</p>
</li>
<li>
<p>From the table of transforms, we can find <span class="process-math">\(x(t)\)</span> as</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
x(t)=e^t + e^{2t} -e^{t/2}\cosh\left(\frac{\sqrt{3}}{2}t\right) -
\frac{1}{\sqrt{3}}e^{t/2}\sinh\left(\frac{\sqrt{3}}{2}t\right).
\end{equation*}
</div>
</li>
<li>
<p>You can find <span class="process-math">\(y(t)\)</span> by differentiating and substituting <span class="process-math">\(\frac{dx}{dt}\)</span> in either of the system equations. Quicker here is to subtract the second equation from the first to obtain</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
-2\frac{dx}{dt}+y+x+2x=e^t-e^{2t}
\end{equation*}
</div>
<p class="continuation">so</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
y(t)=2\frac{dx}{dt}-3x+e^t-e^{2t}.
\end{equation*}
</div>
</li>
</ol></article><span class="incontext"><a href="sec8_4.html#li-77" class="internal">in-context</a></span>
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